
In the following we focus on the conditional probability inside the expectation. Conditional on \(\mathcal {T}_L\) , \(\left(\tau _{L,l,e} - \tau _{L,l,s} \right)^{-1} \left(\sum _{j = \tau _{L,l,s} + 1}^{\tau _{L,l,e}} Y_{nj}\right)^2,\ l = 1,\ldots ,L_{\max }\)  are independent. Bernstein's inequality, see Theorem 2.10 in [1]}, yields that \(\left(\tau _{L,l,e} - \tau _{L,l,s} \right)^{-1/2} \sum _{j = \tau _{L,l,s} + 1}^{\tau _{L,l,e}} Y_{nj}\)  is sub-exponentially distributed with Orlicz norm bounded uniformly over \(l = 1,\ldots ,L_{\max },\ L = 1,\ldots ,L_{\max }\) , i.e. there exist a constant \(\psi ^{(1)} < \infty \)  such that
\(\inf \left\lbrace C \in (0,\infty )\,:\, \mathbb {E}\left[\exp \left( C^{-1}\left(\tau _{L,l,e} - \tau _{L,l,s} \right)^{-1} \left(\sum _{j = \tau _{L,l,s} + 1}^{\tau _{L,l,e}} Y_{nj}\right)^2\right)\right] \le 2 \right\rbrace  \le \psi ^{(1)}\) 
